Proving Weierstrass' Approximation Theorem

Question

I wish to prove that for a continuous function $f$ on a compact subinterval $[a,b]$ of $\mathbb{R}$ and any $\epsilon > 0$, there exists a polynomial $P$ such that

$\sup_{x \in [a,b]} |f(x)-P(x)| < \epsilon$

In the course we have covered the following result: Continuous functions on $\mathbb{T}$ can be uniformly approximated by trigonometric polynomials. I believe this result will be useful in proving the theorem.

Well first I want to assume that $[a,b] = [0,\pi],$ because I can always translate the interval to be $[0,\pi]$. However from here I am unsure how to continue.

Answer 1

Have you heard about "Bernstein polynomials"? These polynomials lead to a very elegant probabilistic proof of Weierstrass approximation theorem. For a proof, see for instance, Proposition 5.2 in the book "Heads or Tails (An Introduction to Limit Theorems in Probability)" written by Emmanuel Lesigne. See also attached snapshot for a proof using Bernstein polynomials.